Mass formulas and stringy point-count for semi-direct products of tame abelian groups and wild symmetric or cyclic groups in characteristic three

TL;DR

This paper explores the failure of Batyrev's theorem in positive characteristic by computing stringy point counts for certain quotient varieties, providing new insights into the McKay correspondence and mass formulas in number theory.

Contribution

It introduces novel computations of stringy point counts for quotient varieties related to semi-direct products of abelian and symmetric or cyclic groups in characteristic three, offering alternative proofs of mass formulas.

Findings

Counterexamples to Batyrev's theorem in positive characteristic identified

Stringy point count computations linked to Serre-Bhagava's mass formula

New variations of mass formula derived from quotient variety analysis

Abstract

In positive characteristic, there exist counterexamples to the statement corresponding to Batyrev's theorem concerning the McKay correspondence. In this paper, we give another computation of the counterexamples by using stringy-point count for some sequences of quotient varieties. There is a proof of the Serre-Bhagava's mass formula, which is important formula in the number theory, from the computation of stringy-point count of a quotient variety associated to a representation of symmetric group. Our computation of the stringy-point count in this paper is regarded as a variation of this proof to give different versions of mass formula.